330 XV NORMED ALGEBRAS AND SPECTRAL THEORY (b) Let 1 <=p < 4- oo, and let/e Jt?p(p,). Show that/B is dense in Jfp(n) if and only if / is an exterior function. (The necessity of the condition follows from Problem 13(a). To show that it is sufficient, observe that if g e &9(p) is such that \gfu dfji = Ofor all u e B,then gfe K(ju), and deduce from Problem 13(h) that# e K(p) and that gfdp, = Q. Apply Problem 13(f) to show that I gdp, = Q and conse- quently that \gu du — 0 for all u e B.) (c) The space -^p(p) is a closed R-submodule of 3ep(p). If B0 is the closed ideal Ker(x) of B, then a closed B-submodule M of &p(p) is said to be proper if B0M is not dense in M. Show that 3^p(p) is a proper B-module. Show that every proper closed B-submodule of &2(p,) is of the form qJ^2(p), where \q\ = 1 . (First remark that there exists in M a nonnegligible function q which is orthogonal to B0M with respect to the hermitian scalar product (/, g) H-> \fg dp,. By expressing that q is orthogonal to #B0 , deduce that (up to a constant factor) we may assume that \q\ = 1 (use Prob- lem 13(j)). Then qJ?2(p) is a closed B-module contained in M. Show that in fact q^2(fji) = M, by remarking that if g e M is orthogonal to q^2(p)y then gq is orthogonal to B, and q is orthogonal to #B0 , hence gq is orthogonal to B0 ; then use Problem 13(k).) Show that the class of q is determined up to a constant factor (use (b)). (d) Deduce from (c) that the Hilbert space L2(/z) is the Hilbert sum of H2(p) (the canonical image of 2^\\^i) and HO(JU) (the canonical image of the closed subspace Jf ?(ju,) consisting of the functions /such that/e^f 2(^) and \fdp, = 0). (Consider the subspace M of &2($ consisting of the functions orthogonal to «^o(/x). Show that M is a proper closed B-module, and therefore of the form q^2(p.) for some function q such that \q\ = 1. Also M contains the constants and hence q e ^f2(/x); conclude that q is equivalent to a constant.) (e) Let/e & *(p,) be such that /does not lie in the closure of/B0 in ^(p). Show that /= gh where h e ^2(p) is an exterior function such that \h\2 =/, and g e &2 lies in the closure of /B in &*(p.). (Apply Problem 12(c) to the function |/|1/2.) If also /e M?l(iJi), then/=#/z where g, h belong to 3V2 and \g\ = \h\. Conversely, if /e ^(JJL) is of the form qh2, where \q\ = 1 and h e ^2(fj) is an exterior function, then /does not belong to the closure of /B0 in JSP^A4) (observe that h2 e £(/*), and use Problem 13 (a)). Equivalently, JM(|/|) > 0. (f) Let M be a proper closed B-submodule of JSfJ(/x). Show that M is of the form qJ^'1(jji)f where \q\ — 1. (Let N = M n -Sf2(ju), which is a closed B-submodule of ^2(ft). If /e M is not in the closure of B0M in &*($, deduce from (e) that/= gh, with g e N and h e «^2(ft); moreover g is not in the closure of B0N in ^2(^), Use (c) to show that N = #.?f2(ju) with \q\ = 1, and deduce that/e^Jf1^)- Applying this result to/H-/i, where/! is in the closure of B0M in ^(p), deduce that/i and hence that M 16. We shall apply the results of Problems 10 to 15 to the Dirichlet algebra B of (15.3,9), where p, is the normalized Haar measure dp,(6) — (277)"1 dO on the unit circle U, and % is the character /(-»/(0), so that p, is the unique representative measure for x (Problem 9(d)). n (a) B is the closure in ^C(U) of the algebra of trigonometric polynomials X) c* ekie all x eK, show that (1 - e)^(ir/K)) < ||f*.||.) 6).pology «^~4, v (use (e)). Give an example of a